Comprehensive nuclear materials 1 11 primary radiation damage formation

Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation Comprehensive nuclear materials 1 11 primary radiation damage formation

R E Stoller

Oak Ridge National Laboratory, Oak Ridge, TN, USA

ß 2012 Elsevier Ltd All rights reserved.

1.11.3 Computational Approach to Simulating Displacement Cascades 297

1.11.4.4 Secondary Factors Influencing Cascade Damage Formation 315

Abbreviations

BCA Binary collision approximation

COM Center of mass

NRT Norgett, Robinson, and Torrens

SIA Self-interstitial atom

Many of the components used in nuclear energy

systems are exposed to high-energy neutrons, which

are a by-product of the energy-producing nuclear

reactions In the case of current fission reactors,

these neutrons are the result of uranium fission,

whereas in future fusion reactors employing rium (D) and tritium (T) as fuel, the neutrons are theresult of DT fusion Spallation neutron sources,which are used for a variety of material researchpurposes, generate neutrons as a result of spallationreactions between a high-energy proton beam and

deute-a hedeute-avy metdeute-al tdeute-arget Neutron exposure cdeute-an ledeute-ad

to substantial changes in the microstructure of thematerials, which are ultimately manifested as observ-able changes in component dimensions and changes

in the material’s physical and mechanical properties

as well For example, radiation-induced void swellingcan lead to density changes greater than 50% in somegrades of austenitic stainless steels1 and changes inthe ductile-to-brittle transition temperature greaterthan 200
C have been observed in the low-alloysteels used in the fabrication of reactor pressurevessels.2,3 These phenomena, along with irradiationcreep and radiation-induced solute segregationare discussed extensively in the literature4 and inmore detail elsewhere in this comprehensive volume(e.g., seeChapter1.03, Radiation-Induced Effects on

293

on Strength and Ductility of Metals and Alloys; and

Chapter1.05, Radiation-Induced Effects on

Mate-rial Properties of Ceramics (Mechanical and

describe the process of primary damage production

that gives rise to macroscopic changes This primary

radiation damage event, which is referred to as an

atomic displacement cascade, was first proposed by

Brinkman in 1954.5,6Many aspects of the cascade

dam-age production discussed below were anticipated in

Brinkman’s conceptual description

In contrast to the time scale required for

radiation-induced mechanical property changes, which is in the

range of hours to years, the primary damage event

that initiates these changes lasts only about 1011s

Similarly, the size scale of displacement cascades,

each one being on the order of a few cubic

nan-ometers, is many orders of magnitude smaller than

the large structural components that they affect

Although interest in displacement cascades was

initi-ally limited to the nuclear industry, cascade damage

production has become important in the solid state

processing practices of the electronics industry also.7

The cascades of interest to the electronics industry

arise from the use of ion beams to fabricate, modify,

or analyze materials for electronic devices Another

related application is the modification of surface

layers by ion beam implantation to improve wear or

corrosion resistance of materials.8 The energy and

mass of the particle that initiates the cascade provide

the principal differences between the nuclear and

ion beam applications Neutrons from nuclear fission

and DT fusion have energies up to about 20 MeV and

14.1 MeV, respectively, while the peak neutron

energy in spallation neutron sources reaches as high

as the energy of the incident proton beam,1 GeV in

modern sources.9 The neutron mass of one atomic

mass unit (1 a.m.u.1.66  10–27

kg) is much less thanthat of the mid-atomic weight metals that comprise

most structural alloys In contrast, many ion beam

applications involve relatively low-energy ions, a few

tens of kiloelectronvolts, and the mass of both the

incident particle and the target is typically a few tens

of atomic mass unit The use of somewhat higher

energy ion beams as a tool for investigating neutron

irradiation effects is discussed in Chapter 1.07,

This chapter will focus on the cascade energies of

relevance to nuclear energy systems and on iron,

which is the primary component in most of the alloys

employed in these systems However, the description

of the basic physical mechanisms of displacement

cascade formation and evolution given below is erally valid for any crystalline metal and for all of theapplications mentioned above Although additionalphysical processes may come into play to alter thefinal defect state in ionic or covalent materials due toatomic charge states,10the ballistic processes observed

gen-in metals due to displacement cascades are quitesimilar in these materials This has been demon-strated in molecular dynamics (MD) simulations in

a range of ceramic materials.11–15Finally, synergisticeffects due to nuclear transmutation reactions willnot be addressed; the most notable of these, heliumproduction by (n,a) reactions, is the topic of Chapter1.06, The Effects of Helium in Irradiated Struc-

Cascades

In a crystalline material, a displacement cascade can

be visualized as a series of elastic collisions that isinitiated when a given atom is struck by a high-energyneutron (or incident ion in the case of ion irradiation).The initial atom, which is called the primary knock-onatom (PKA), will recoil with a given amount of kineticenergy that it dissipates in a sequence of collisionswith other atoms The first of these are termed sec-ondary knock-on atoms and they will in turn loseenergy to a third and subsequently higher orderedknock-ons until all of the energy initially imparted

to the PKA has been dissipated Although the physics

is slightly different, a similar event has been observed

on billiard tables for many years

Perhaps the most important difference betweenbilliards and atomic displacement cascades is that

an atom in a crystalline solid experiences the bindingforces that arise from the presence of the other atoms.This binding leads to the formation of the crystallinelattice and the requirement that a certain minimumkinetic energy must be transferred to an atom before

it can be displaced from its lattice site This minimumenergy is called the displacement threshold energy(Ed) and is typically 20 to 40 eV for most metals andalloys used in structural applications.16

If an atom receives kinetic energy in excess ofEd,

it can be transported from its original lattice site andcome to rest within the interstices of the lattice Such

an atom constitutes a point defect in the lattice and iscalled an interstitial or interstitial atom In the case of

an alloy, the interstitial atom may be referred to as aself-interstitial atom (SIA) if the atom is the primary

alloy component (e.g., iron in steel) to distinguish it

from impurity or solute interstitials The SIA

nomen-clature is also used for pure metals, although it is

somewhat redundant in that case The

complemen-tary point defect is formed if the original lattice site

remains vacant; such a site is called a vacancy (see

Chapter1.01, Fundamental Properties of Defects

properties) Vacancies and interstitials are created in

equal numbers by this process and the name Frenkel

pair is used to describe a single, stable interstitial

and its related vacancy Small clusters of both point

defect types can also be formed within a

displace-ment cascade

The kinematics of the displacement cascade

can be described as follows, where for simplicity we

consider the case of nonrelativistic particle energies

with one particle initially in motion with kinetic

energyE0and the other at rest In an elastic collision

between two such particles, the maximum energy

transfer (Em) from particle (1) to particle (2) is given by

Em¼ 4EoA1A2=ðA1þ A2Þ2 ½1

where A1 and A2 are the atomic masses of the two

particles Two limiting cases are of interest If particle

1 is a neutron and particle 2 is a relatively heavy

element such as iron, Em 4E0/A Alternately, if

A1¼ A2, any energy up toE0can be transferred The

former case corresponds to the initial collision between

a neutron and the PKA, while the latter corresponds

to the collisions between lattice atoms of the same mass

Beginning with the work of Brinkman mentioned

above, various models were proposed to compute

the total number of atoms displaced by a given PKA

as a function of energy The most widely cited

model was that of Kinchin and Pease.17Their model

assumed that between a specified threshold energy

and an upper energy cut-off, there was a linear

rela-tionship between the number of Frenkel pair

pro-duced and the PKA energy Below the threshold, no

new displacements would be produced Above the

high-energy cut-off, it was assumed that the

addi-tional energy was dissipated in electronic excitation

and ionization Later, Lindhard and coworkers

devel-oped a detailed theory for energy partitioning that

could be used to compute the fraction of the PKA

energy that was dissipated in the nuclear system in

elastic collisions and in electronic losses.18This work

was used by Norgett, Robinson, and Torrens (NRT)

to develop a secondary displacement model that is

still used as a standard in the nuclear industry and

elsewhere to compute atomic displacement rates.19

The NRT model gives the total number of placed atoms produced by a PKA with kinetic energy

An example of the angular dependence is shown in

the Finnis–Sinclair potential.21 Moreover, it is notobvious how to obtain a unique definition for theangular average Nordlund and coworkers22provide acomparison of threshold behavior obtained with 11different iron potentials and discusses several differentpossible definitions of the displacement thresholdenergy The factorTdineqn [2]is called the damageenergy and is a function ofEPKA The damage energy isthe amount of the initial PKA energy available to causeatomic displacements, with the fraction of the PKA’sinitial kinetic energy lost to electronic excitationbeing responsible for the difference between EPKA

and Td The ratio of Td to EPKA for iron is shown

the analytical fit to Lindhard’s theory described byNorgett and coworkers19has been used to obtainTd.Note that a significant fraction of the PKA energy

is dissipated in electronic processes even for energies

Figure 1 Angular dependence of displacement threshold energy for iron at 0 K Reproduced from Bacon, D J.; Calder, A F.; Harder, J M.; Wooding, S J J Nucl Mater.

1993, 205, 52–58.

as low as a few kiloelectronvolts The factor of

0.8 in eqn [2] accounts for the effects of realistic

(i.e., other than hard sphere) atomic scattering; the

value was obtained from an extensive cascade study

using the binary collision approximation (BCA).23,24

The number of stable displacements (Frenkel

pair) predicted by both the original Kinchin–Pease

model and the NRT model is shown inFigure 3as

a function of the PKA energy The third curve in

the figure will be discussed below inSection 1.11.3

The MD results presented inSection 1.11.4.2

indi-cate that nNRT overestimates the total number of

Frenkel pair that remain after the excess kineticenergy in a displacement cascade has been dis-sipated at about 10 ps Many more defects thanthis are formed during the collisional phase of thecascade; however, most of these disappear as vacan-cies and interstitials annihilate one another in spon-taneous recombination reactions

One valuable aspect of the NRT model is that itenabled the use of atomic displacements per atom(dpa) as an exposure parameter, which provides acommon basis of comparison for data obtained indifferent types of irradiation sources, for example,different neutron energy spectra, ion irradiation,

or electron irradiation The neutron energy trum can vary significantly from one reactor toanother depending on the reactor coolant and/ormoderator (water, heavy water, sodium, graphite),which leads to differences in the PKA energy spec-trum as will be discussed below This can confoundattempts to correlate irradiation effects data on thebasis of parameters such as total neutron fluence orthe fluence above some threshold energy, commonly0.1 or 1.0 MeV More importantly, it is impossible tocorrelate any given neutron fluence with a chargedparticle fluence However, in any of these cases, thePKA energy spectrum and corresponding damageenergies can be calculated and the total number ofdisplacements obtained using eqn [2] in an integralcalculation Thus, dpa provides an environment-independent radiation exposure parameter that in

spec-14

12 10

8

6

4 2

0

1200 1000 800 600 400 200 0

0.8 PKA energy (keV)

PKA energy (keV)

Figure 3 Predicted Frenkel pair production as a function of PKA energy for alternate displacement models (see text for explanation of models).

PKA energy (keV)

Figure 2 Ratio of damage energy (T d ) to PKA energy

(E PKA ) as a function of PKA energy.

many cases can be successfully used as a radiation

damage correlation parameter.25 As discussed below,

aspects of primary damage production other than

simply the total number of displacements must be

considered in some cases

Simulating Displacement Cascades

Given the short time scale and small volume associated

with atomic displacement cascades, it is not currently

possible to directly observe their behavior by any

avail-able experimental method Some of their characteristics

have been inferred by experimental techniques that can

examine the fine microstructural features that form after

low doses of irradiation The experimental work that

provides the best estimate of stable Frenkel pair

produc-tion involves cryogenic irradiaproduc-tion and subsequent

annealing while measuring a parameter such as

electri-cal resistivity.26,27 Less direct experimental

measure-ments include small angle neutron scattering,28X-ray

scattering,29 positron annihilation spectroscopy,30 and

field ion microscopy.31More broadly, transmission

elec-tron microscopy (TEM) has been used to characterize

the small point defect clusters such as microvoids,

dislocation loops, and stacking fault tetrahedra that

are formed as the cascade collapses.32–36

The primary tool for investigating radiation

dam-age formation in displacement cascades has been

computer simulation using MD, which is a

computa-tionally intensive method for modeling atomic

sys-tems on the time and length scales appropriate to

displacement cascades The method was pioneered

by Vineyard and coworkers at Brookhaven National

Laboratory,37and much of the early work on atomistic

simulations is collected in a review by Beeler.38Other

methods, such as those based on the BCA,20,21 have

also been used to study displacement cascades The

binary collision models are well suited for very

high-energy events, which require that the interatomic

potential accurately simulate only close encounters

between pairs of atoms This method requires

sub-stantially less computer time than MD but provides

less detailed information about lower energy

colli-sions where many-body effects become important

In addition, in-cascade recombination and clustering

can only be treated parametrically in the BCA When

the necessary parameters have been calibrated using

the results of an appropriate database of MD cascade

results, the BCA codes have been shown to reproduce

the results of MD simulations reasonably well.39,40

A detailed description of the MD method isgiven in Chapter 1.09, Molecular Dynamics, andwill not be repeated here Briefly, the method relies onobtaining a sufficiently accurate analytical inter-atomic potential function that describes the energy

of the atomic system and the forces on each atom as afunction of its position relative to the other atoms

in the system This function must account for bothattractive and repulsive forces to obtain the appropri-ate stable lattice configuration Specific values for theadjustable coefficients in the function are obtained

by ensuring that the interatomic potential leads toreasonable agreement with measured material para-meters such as the lattice parameter, lattice cohesiveenergy, single crystal elastic constants, melting tem-perature, and point defect formation energies Theprocess of developing and fitting interatomic poten-tials is the subject of Chapter 1.10, Interatomic

when using MD and an empirical potential to gate radiation damage, viz the distance of closestapproach for highly energetic atoms is much smallerthan that obtained in any equilibrium condition Mostpotentials are developed to describe equilibrium con-ditions and must be modified or ‘stiffened’ to accountfor these short-range interactions Chapter 1.10,

common approach in which a screened Coulombpotential is joined to the equilibrium potential forthis purpose However, as Malerba points out,41criti-cal aspects of cascade behavior can be sensitive to thedetails of this joining process

When this interatomic potential has been derived,the total energy of the system of atoms being simulatedcan be calculated by summing over all the atoms Theforces on the atoms are obtained from the gradient ofthe interatomic potential These forces can be used

to calculate the atom’s accelerations according toNewton’s second law, the familiar F ¼ ma (force ¼mass acceleration), and the equations of motionfor the atoms can be solved by numerical integrationusing a suitably small time step At the end of thetime step, the forces are recalculated for the newatomic positions and this process is repeated as long

as necessary to reach the time or state of interest.For energetic PKA, the initial time step may rangefrom1 to 10  1018s, with the maximum time steplimited to 1–10  1015s to maintain acceptablenumerical accuracy in the integration As a result,

MD cascade simulations are typically not run fortimes longer than 10–100 ps With periodic boundaryconditions, the size of the simulation cell needs to be

large enough to prevent the cascade from interacting

with periodic images of itself Higher energy events

therefore require a larger number of atoms in the

cell Typical MD cascade energies and the

approxi-mate number of atoms required in the simulation

are listed inTable 1 With periodic boundaries, it is

important that the cell size be large enough to avoid

cascade self-interaction For a given energy, this

size depends on the material and, for a given material,

on the interatomic potential used Different

inter-atomic potentials may predict significantly different

cascade volumes, even though little variation is

even-tually found in the number of stable Frenkel pair.42

Using a modest number of processors on a modern

parallel computer, the clock time required to

com-plete a high-energy simulation with several million

atoms is generally less than 48 h Longer-term

evolu-tion of the cascade-produced defect structure can be

carried out using Monte Carlo (MC) methods as

discussed in Chapter 1.14, Kinetic Monte Carlo

The process of conducting a cascade simulation

requires two steps First, a block of atoms of the

desired size is thermally equilibrated This permits

the lattice thermal vibrations (phonon waves) to be

established for the simulated temperature and

typi-cally requires a simulation time of approximately

10 ps This equilibrated atom block can be saved and

used as the starting point for several subsequent

cas-cade simulations Subsequently, the cascas-cade

simula-tions are initiated by giving one of the atoms a defined

amount of kinetic energy,EMD, in a specified

direc-tion Statistical variability can be introduced by either

further equilibration of the starting block, choosing adifferent PKA or PKA direction, or some combination

of these The number of simulations required at anyone condition to obtain a good statistical description

of defect production is not large Typically, only about8–10 simulations are required to obtain a small stan-dard error about the mean number of defects pro-duced; the scatter in defect clustering parameters islarger This topic will be discussed further belowwhen the results are presented Most of the cascadesimulations discussed below were generated using a[135] PKA direction to minimize directional effectssuch as channeling and directions with particularly low

or high displacement thresholds The objective hasbeen to determine mean behavior, and investigations

of the effect of PKA direction generally indicate thatmean values obtained from [135] cascades are repre-sentative of the average defect production expected incascades greater than about 1 keV.43A stronger influ-ence of PKA direction can be observed at lower ener-gies as discussed in Stoller and coworkers.44,45

In the course of the simulation, some proceduremust be applied to determine which of the atomsshould be characterized as being in a defect statefor the purpose of visualization and analysis Oneapproach is to search the volume of a Wigner–Seitzcell, which is centered on one of the original, perfectlattice sites An empty cell indicates the presence of avacancy and a cell containing more than one atomindicates an interstitial-type defect A more simplegeometric criterion has been used to identify defects

in most of the results presented below A sphere with

a radius equal to 30% of the iron lattice parameter is

Table 1 Typical iron atomic displacement cascade parameters

Neutron energy

(MeV)

Average PKA energy (keV) a

Corresponding

T d (keV) b

E MD

NRT displacements

Ratio:

T d /E PKA

Typical simulation cell size (atoms)

aThis is the average iron recoil energy from an elastic collision with a neutron of the specified energy.

bDamage energy calculated using Robinson’s approximation to LSS theory.19

c

centered on the perfect lattice sites, and a search

similar to that just described for the Wigner–Seitz

cell is carried out Any atom that is not within such

a sphere is identified as part of an interstitial defect

and each empty sphere identifies the location of

a vacancy The diameter of the effective sphere

is slightly less than the spacing of the two atoms

in a dumbbell interstitial (see below) A comparison

of the effective sphere and Wigner–Seitz cell

approaches found no significant difference in the

number of stable point defects identified at the end

of cascade simulation, and the effective sphere

method is faster computationally The drawback to

this approach is that the number of defects

identi-fied by the algorithm must be corrected to account

for the nature of the interstitial defect that is

formed In order to minimize the lattice strain

energy, most interstitials are found in the dumbbell

configuration; the energy is reduced by distributing

the distortion over multiple lattice sites In this case,

the single interstitial appears to be composed of two

interstitials separated by a vacancy In other cases,

the interstitial configuration is extended further, as in

the case of the crowdion in which an interstitial may

be visualized as three displaced atoms and two

empty lattice sites These interstitial configurations

are illustrated inFigure 4, which uses the convention

adopted throughout this chapter, that is, vacancies are

displayed as red spheres and interstitials as green

spheres A simple postprocessing code was used to

determine the true number of point defects, which

are reported below

Most MD codes describe only the elastic

colli-sions between atoms; they do not account for energy

loss mechanisms such as electronic excitation andionization Thus, the initial kinetic energy, EMD,given to the simulated PKA in MD simulations ismore analogous to Td in eqn [2] than it is to thePKA energy, which is the total kinetic energy of therecoil in an actual collision Using the values ofEMD

nNRTfor iron, and the ratio of the damage energy tothe PKA energy, have been calculated using theprocedure described in Norgett and coworkers.19and the recommended 40 eV displacement thresh-old.16These values are also listed in Table 1, alongwith the neutron energy that would yieldEPKAas theaverage recoil energy in iron This is one-half of themaximum energy given by eqn [1] As mentionedabove, the difference between the MD cascade energy,

or damage energy, and the PKA energy increases asthe PKA energy increases Discussions of cascadeenergy in the literature on MD cascade simulationsare not consistent with respect to the use of the termPKA energy The third curve inFigure 3shows thecalculated number of Frenkel pair predicted by theNRT model if the PKA energy is used in eqn [2]rather than the damage energy The differencebetween the two sets of NRT values is substantialand is a measure of the ambiguity associated withbeing vague in the use of terminology It is recom-mended that the MD cascade energy should not bereferred to as the PKA energy For the purpose ofcomparing MD results to the NRT model, the MDcascade energy should be considered as approxi-mately equal to the damage energy (Tdineqn [2])

In reality, energetic atoms lose energy ously by a combination of electronic and nuclearreactions, and the typical MD simulation effectivelydeletes the electronic component at time zero Theeffects of continuous energy loss on defect produc-tion have been investigated in the past using a damp-ing term to slowly remove kinetic energy.46 Therelated issues of how this extracted energy heats theelectron system and the effects of electron–phononcoupling on local temperature have also been exam-ined.47–50More recently, computational and algorith-mic advances have enabled these phenomena to beinvestigated with higher fidelity.51Some of the workjust referenced has shown that accounting for theelectronic system has a modest quantitative effect ondefect formation in displacement cascades For exam-ple, Gao and coworkers found a systematic increase

continu-in defect formation as they continu-increased the effectiveelectron–phonon coupling in 2, 5, and 10 keV cascadesimulations in iron,50and a similar effect was reported

z

[111]

[110]

[111]

Figure 4 Typical configurations for interstitials created in

displacement cascades: [110] and [111] dumbbells and

[111] crowdion.

by Finnis and coworkers.47 However, the primary

physical mechanisms of defect formation that are

the focus of this chapter can be understood in the

absence of these effects

Simulations in Iron

MD simulations have been employed to investigate

displacement cascade evolution in a wide range of

materials The literature is sufficiently broad that

any list of references will be necessarily incomplete;

Malerba,41 Stoller,43 and others52–70 provide only a

representative sample Additional references will be

given below as specific topics are discussed The

recent review by Malerba41provides a good summary

of the research that has been done on iron These MD

investigations of displacement cascades have

estab-lished several consistent trends in primary damage

formation in a number of materials These trends

include (1) the total number of stable point defects

produced follows a power-law dependence on the

cas-cade energy over a broad energy range, (2) the ratio

of MD stable displacements divided by the number

obtained from the NRT model decreases with energy

until subcascade formation becomes prominent,

(3) the in-cascade clustering fraction of the surviving

defects increases with cascade energy, and (4) the

effect of lattice temperature on the MD results is

rather weak Two additional observations have been

made regarding in-cascade clustering in iron, although

the fidelity of these statements depends on the

interatomic potential employed First, the interstitial

clusters have a complex, three-dimensional (3D)

morphology, with both sessile and glissile

configura-tions Mobile interstitial clusters appear to glide with

a low activation energy similar to that of the

mono-interstitial (0.1–0.2 eV).71

Second, the fraction of thevacancies contained in clusters is much lower than

the interstitial clustering fraction Each of these points

will be discussed further below

The influence of the interatomic potential on

cascade damage production has been investigated by

several researchers.42,72–74Such comparisons generally

show only minor quantitative differences between

results obtained with interatomic potentials of the

same general type, although the differences in

cluster-ing behavior are more significant with some potentials

Variants of embedded atom or Finnis–Sinclair type

potential functions (see Chapter 1.10, Interatomic

However, more substantial differences are sometimesobserved that are difficult to correlate with any knownaspect of the potentials The analysis recently reported

by Malerba41 is one example In this case, it appearsthat the formation of replacement collision sequences(RCS) (discussed inSection 1.11.4.1) was very sensi-tive to the range over which the equilibrium part ofthe potential was joined to the more repulsive pairpotential that controls short-range interactions Thischanged the effective cascade energy density andthereby the number of stable defects produced.Therefore, in order to provide a self-consistentdatabase for illustrating cascade damage productionover a range of temperatures and energies and toprovide examples of secondary variables that caninfluence this production, the results presented inthis chapter will focus on MD simulations in ironusing a single interatomic potential.43,53,54,64–68Thispotential was originally developed by Finnis andSinclair21and later modified for cascade simulations

by Calder and Bacon.58The calculations were carriedout using a modified version of the MOLDY codewritten by Finnis.75 The computing time with thiscode is almost linearly proportional to the number ofatoms in the simulation Simulations were carried outusing periodic, Parrinello–Rahman boundary condi-tions at constant pressure.76 As no thermostat wasapplied to the boundaries, the average temperature

of the simulation cell was increased as the kineticenergy of the PKA was dissipated The impact ofthis heating appears to be modest based on theobserved effects of irradiation temperature discussedbelow, and on the results observed in the work of Gaoand coworkers.77A brief comparison of the iron cas-cade results with those obtained in other metals will

be presented inSection 1.11.5.The primary variables studied in these cascadesimulations is the cascade energy,EMD, and the irra-diation temperature The database of iron cascadesincludes cascade energies from near the displacementthreshold (100 eV) to a 200 keV, and temperatures

in the range of 100–900 K In all cases, the evolution

of the cascade has been followed to completion andthe final defect state determined Typically this isreached after a few picoseconds for the low-energycascades and up to 15 ps for the highest energycascades Because of the variability in final defectproduction for similar initial conditions, several simu-lations were conducted at each energy to producestatistically meaningful average values The para-meters of most interest from these studies are thenumber of surviving point defects, the fraction of

these defects that are found in clusters, and the size

distribution of the point defect clusters The total

number of point defects is a direct measure of the

residual radiation damage and the potential for

long-range mass transport and microstructural evolution

In-cascade defect clustering is important because it

can promote microstructural evolution by

eliminat-ing the cluster nucleation phase

The parameters used in the following discussion

to describe results of MD cascade simulations are

the total number of surviving point defects and the

fraction of the surviving defects contained in

clus-ters The number of surviving defects will be

expressed as a fraction of the NRT displacements

listed inTable 1, whereas the number of defects in

clusters will be expressed as either a fraction of the

NRT displacements or a fraction of the total

surviv-ing MD defects Alternate criteria were used to

define a point defect cluster in this study In the

case of interstitial clusters, it was usually

deter-mined by direct visualization of the defect

struc-tures The coordinated movement of interstitials in a

given cluster can be clearly observed Interstitials

bound in a given cluster were typically within asecond nearest-neighbor (NN) distance, althoughsome were bound at third NN The situation forvacancy clusters will be discussed further below,but vacancy clustering was assessed using first, sec-ond, third, and fourth NN distances as the criteria.The vacancy clusters observed in iron tend to notexhibit a compact structure according to these defi-nitions In order to analyze the statistical variation inthe primary damage parameters, the mean value(M), the standard deviation about the mean (s),and the standard error of the mean (e) have beencalculated for each set of cascades conducted at agiven energy and temperature The standard error

of the mean is calculated ase ¼ s/n0.5

, wheren is thenumber of cascade simulations completed.78 Thestandard error of the mean provides a measure ofhow well the sample mean represents the actualmean For example, a 90% confidence limit on themean is obtained from 1.86e for a sample size ofnine.79These statistical quantities are summarized

cascade database

Table 2 Statistical analysis of primary damage parameters derived from MD cascade simulations

Energy (keV) Temperature (K) Number of cascades Surviving MD

displacements (mean / standard deviation / standard error)

Clustered interstitials (mean / standard deviation / standard error)

Number per NRT Number per NRT per MD

surviving defects 3.94 0.790 1.25 0.250 0.310

0.170 0.0340 0.348 0.0695 0.0822 6.08 0.608 2.25 0.225 0.341

0.398 0.0398 0.479 0.0479 0.0715 5.25 0.525 1.92 0.192 0.307

0.579 0579 0.583 0.0583 0.0944 4.33 0.433 1.00 0.100 0.221

0.310 0.031 0.369 0.0369 0.0829 10.1 0.505 4.60 0.230 0.432

0.836 0.0418 0.884 0.0442 0.00678 22.0 0.440 11.4 0.229 0.523

0.707 0.0141 0.801 0.0160 0.0375

Continued

Table 2 Continued

Energy (keV) Temperature (K) Number of cascades Surviving MD

displacements (mean / standard deviation / standard error)

Clustered interstitials (mean / standard deviation / standard error)

Number per NRT Number per NRT per MD

surviving defects 19.1 0.382 9.77 0.195 0.504

1.08 0.0215 1.13 0.0227 0.0520 17.1 0.343 8.38 0.168 0.488

0.915 0.0183 0.653 0.0131 0.0261 33.6 0.336 17.0 0.170 0.506

1.37 0.0137 1.04 0.0104 0.0261 30.5 0.305 18.1 0.181 0.579

3.66 0.0366 2.99 0.0299 0.0406 27.3 0.273 18.6 0.186 0.679

2.14 0.0214 2.29 0.0229 0.00606 60.2 0.301 36.7 0.184 0.610

2.76 0.0138 2.06 0.0103 0.0199 55.8 0.281 41.6 0.211 0.746

2.09 0.0103 2.07 0.0101 0.0281 51.7 0.259 35.4 0.177 0.682

3.09 0.0154 2.83 0.0141 0.0299 94.9 0.316 57.2 0.191 0.602

3.29 0.0110 2.88 0.00963 0.0209 131.0 0.328 74.5 0.186 0.570

4.45 0.0111 5.30 0.0133 0.0361 168.3 0.337 93.6 0.187 0.557

4.04 0.00807 2.32 0.00463 0.0144 329.7 0.330 184.8 0.185 0.561

8.93 0.0089 6.47 0.00650 0.0122 282.4 0.282 185.5 0.186 0.656

5.95 0.00595 6.01 0.00601 0.0124 261.0 0.261 168.7 0.169 0.646

4.13 0.00413 4.08 0.00408 0.0117 676.7 0.338 370.3 0.185 0.548

12.6 0.00632 9.83 0.00491 0.0155

1.11.4.1 Cascade Evolution and Structure

The evolution of displacement cascades is similar at

all energies, with the development of a highly

ener-getic, disordered core region during the initial,

colli-sional phase of the cascade Vacancies and interstitials

are created in equal numbers, and the number of

point defects increases sharply until a peak value is

reached Depending on the cascade energy, this

occurs at a time in the range of 0.1–1 ps This

evolu-tion is illustrated inFigure 5for a range of cascade

energies, where the number of vacancies is shown

as a function of the cascade time Many vacancy–

interstitial pairs are in quite close proximity at the

time of peak disorder An essentially athermal process

of in-cascade recombination of these close pairs takes

place as they lose their kinetic energy This leads to

a reduction in the number of defects until a

quasi-steady-state value is reached after about 5–10 ps

As interstitials in iron are mobile even at 100 K,

further short-term recombination occurs between

some vacancy-interstitial pairs that were initially

separated by only a few atomic jump distances Finally,

a stage is reached where the remaining point defects

are sufficiently well separated that further

recombina-tion is unlikely on the time scale (a few hundred

picoseconds) accessible by MD Note that the number

of stable Frenkel pair is actually somewhat lower

than the value shown inFigure 5because the values

obtained using the effective sphere identification

procedure were not corrected to account for theinterstitial structure discussed above

A mechanism known as RCS may help explainsome aspects of cascade structure.24,41 An RCS can

be visualized as an extended defect along a packed row of atoms When the first atom is pushedoff its site, it dissipates some energy and pushes asecond atom into a third, and so on When the lastatom in this chain is unable to displace another, it isleft in an interstitial site with the original vacancyseveral atomic jumps away Thus, RCSs provide amechanism of mass transport that can efficiently sep-arate vacancies from interstitials The explanation isconsistent with the observed tendency for the finalcascade state to be characterized by a vacancy-richcentral region that is surrounded by a region rich

close-in close-interstitial-type defects However, although RCSsare observed, particularly in low-energy cascades,they do not appear to be prominent enough to explainthe defect separation observed in higher energycascades.58Visualization of cascade dynamics indicatesthat the separation occurs by a more collective motion

of multiple atoms, and recent work by Calder andcoworkers has identified a shockwave-induced mech-anism that leads to the formation of large interstitialclusters at the cascade periphery.80This mechanismwill be discussed further in Section 1.11.4.3.1.Coherent displacement events involving many atomshave also been reported by Nordlund and coworkers.81

Defect production tends to be dominated by a

series of simple binary collisions at low PKA

ener-gies, while the more collective, cascade-like

behav-ior dominates at higher energies The structure of

typical 1 and 20 keV cascades is shown inFigure 6,

where parts (a) and (b) show the peak damage state

and (c) and (d) show the final defect configurations

The MD cells contained 54 000 and 432 000 atoms

for the 1 and 20 keV simulations, respectively Only

the vacant lattice sites and interstitial atoms

identi-fied by the effective sphere approach described

above are shown The separation of vacancies from

interstitials can be seen in the final defect

config-urations; it is more obvious in the 1 keV cascade

because there are fewer defects present In addition

to isolated point defects, small interstitial clusters

are also clearly observed in the 20 keV cascade

debris inFigure 6(d) In-cascade clustering is

dis-cussed further inSection 1.11.4.3

The morphology of the 20 keV cascade in

of a phenomenon known as subcascade formation.82

At low energies, the PKA energy tends to be

dis-sipated in a small volume and the cascades appear as

compact, sphere-like entities as illustrated by the

1 keV cascade in Figure 6(a) However, at higherenergies, some channeling82,83of recoil atoms mayoccur This is a result of the atom being scatteredinto a relatively open lattice direction, which maypermit it to travel some distance while losing rela-tively little energy in low-angle scattering events.The channeling is typically terminated in a high-angle collision in which a significant fraction ofthe recoil atom’s energy is transmitted to the nextgeneration knock-on atom When significant sub-cascade formation occurs, the region betweenhigh-angle collisions can be relatively defect-free asthe cascade develops This evolution is clearly shown

due to high-angle collisions is observed on a time scale

of a few hundreds of femto seconds One practicalimplication of subcascade formation is that veryhigh-energy cascades break up into what looks like agroup of lower energy cascades An example of sub-cascade formation in a 100 keV cascade is shown in

have been superimposed into the same block of atomsfor comparison The impact of subcascade formation

on stable defect production will be discussed in thenext section

x x

z

z

z

y y

Vacancy Interstitial

Figure 6 Structure of typical 1 keV (a,c) and 20 keV (b,d) cascades Peak damage state is shown in (a and b) and the final stable defect configuration is shown in (c and d).

1.11.4.2 Stable Defect Formation

Initial work of Bacon and coworkers indicated that

the number of stable displacements remaining at the

end of a cascade simulation,ND, exhibited a

power-law dependence on cascade energy.84 For example,

their analysis of iron cascade simulations between 0.5

and 10 keV at 100 K showed that the total number of

surviving point defects could be expressed as

ND¼ 5:67E0:779

where EMD is given in kiloelectronvolts This

rela-tionship is not followed below about 0.5 keV because

true cascade-like behavior does not occur at these

low energies Subsequent work by Stoller64–67cated thatNDalso begins to deviate from this energydependence above 20 keV when extensive subcascadeformation occurs This is illustrated in Figure 9(a)where the values ofNDobtained in cascade simula-tions at 100 K is plotted as a function of cascadeenergy At each energy, the data point is an average

indi-of between 7 and 26 cascades, and the error barsindicate the standard error of the mean It appearsthat three well-defined regions with different energydependencies exist A power-law fit to the points

in each energy region is also shown in Figure 9(a).The best-fit exponent in the absence of true cas-cade conditions below 0.5 keV is 0.485 From 0.5 to

Figure 7 Evolution of a 40 keV cascade in iron at 100 K, illustrating subcascade formation.

MD cascade simulations in iron at 100 K: peak damage

20 keV, the exponent is 0.75 This is marginally

lower than the value in eqn [2], possibly because

the 20 keV data were used in the current fitting

An exponent of 1.03 was found in the range above

20 keV, which is dominated by subcascade formation

Only in the highest energy range do the MD results

approach the linear energy dependence predicted

by the NRT model The range of plus or minus

one standard error is barely detectable around the

data points, indicating that the change in slope is

statistically significant

The data from Figure 9(a) are replotted in

displace-ments is divided by the NRT displacedisplace-ments at each

energy The rapid decrease in this MD defect survival

ratio at low energies was first measured in 1978 and

is well known.57,85 The error bars again reflect thestandard error and the dashed line through the points

is only a guide to the eye The MD/NRT ratio isgreater than 1.0 at the lowest values ofEMD, indicat-ing that the NRT formulation underestimates defectproduction in this energy range This is consistentwith the low-energy (near threshold) simulationspreferentially producing displacements in the ‘easy’directions.26The actual displacement threshold var-ies with crystallographic direction and is as low

as 19 eV in the [100] direction.20,84

Thus, usingthe recommended average value of 40 eV Edineqn[2]predicts fewer defects at low energies The aver-age value is more appropriate for the higher energyevents where true cascade-like behavior occurs Inthe cascade-dominated regime, the defect densitywithin the cascade increases with energy Althoughmany more defects are produced, their close proxim-ity leads to a higher probability of in-cascade recom-bination and a lower defect survival fraction.The surviving defect fraction shows a slightincrease as the cascade energy increases above 20,and the indicated standard errors make it arguablethat the increase is statistically significant If signifi-cant, the increase appears to be associated with sub-cascade formation, which becomes prominent above10–20 keV In the channeling regions between thehigh-angle collisions that produce the subcascadesshown in Figures 7 and 8, the moving atom losesenergy in many low-angle scattering events that pro-duce low-energy recoils These are essentially likelow-energy cascades, which have higher-than-averagedefect survival fractions (Figure 9) These events couldcontribute to the incremental increase in defect sur-vival at the highest energies The average defect sur-vival fraction of0.3 NRT shown for cascade energiesgreater than about 10 keV is consistent with values ofFrenkel pair formation obtained from resistivitychange measurements following low-temperatureneutron irradiation and ion irradiation.26,27,57,85The effect of irradiation temperature is shown

fractions obtained from simulations at 100, 600, and

900 K Although it is difficult to discern a consistenteffect of temperature between the 600 and

900 K data points, the defect survival fraction at

100 K is always somewhat greater than at either

of the two higher temperatures A similar resultfor iron was reported in Bacon and coworkers.84Inaddition to an interest in radiation temperatureitself, the effect of temperature is relevant to the

Cascade energy (keV)

Figure 9 Cascade energy dependence of stable point

defect formation in iron MD cascade simulations at 100 K:

(a) total number of interstitials or vacancies and (b) ratio of

MD defects to NRT displacements Data points indicate

mean values at each energy, and error bars are standard

error of the mean.

simulations presented here because no thermostat

was applied to the simulation cell to control

tem-perature As mentioned above, the energy

intro-duced by the PKA will lead to some heating if the

simulation cell temperature is not controlled by a

thermostat For example, in a 1 keV cascade

simula-tion with 54 000 atoms, the average temperature rise

will be about 140 K when all the kinetic energy

of the PKA is distributed in the system This change

in temperature should be more significant at

100 K than at higher temperatures The fact that

defect survival at 600 and 900 K is lower than at

100 K suggests that the 100 K results may be

somewhat biased toward lower survival values bythe PKA-induced heating This is in agreementwith the effect of temperature reported by Gao andcoworkers77in their study of 2 and 5 keV cascades with

a hybrid MD model that extracted heat from the lation cell On the other hand, the difference betweenthe 100 and 600 K results is not large, so the effect of

simu-200 K of cascade-induced heating may be modest

A simple assessment of this cascade-induced ing was carried out using 10 keV cascades at 100 K.Two independent sets of simulations were carried out,seven simulations in a cell of 128 k atoms and eightsimulations in a cell of 250 k atoms A 10 keV cascadewill raise the average temperature by 604 and 309 K,respectively, for these two cell sizes The results ofthese simulations are summarized in Figure 11,where the parameters plotted are the surviving defectfraction (per NRT), the fraction of interstitials inclusters (per NRT), and the fraction of interstitials

heat-in clusters (per survivheat-ing MD defect) In each case,the range of values for the two populations are shown,along with their respective mean values with thestandard error indicated The mean and standarderror for the combined data sets is also shown.Although the heating differed by a factor of two, it isclear that the defect survival fraction is essentiallyidentical for both populations There is a slight trend

in the interstitial clustering results, which indicatesthat a higher temperature (due to a smaller number ofatoms) promotes interstitial clustering This is consis-tent with the results that will be discussed below

0.7

Averages and standard error for indicated data points 0.5

0.4

0.3

0.2 0.1

7–10 keV in 128 k atom 8–10 keV in 250 k atoms

Figure 10 Temperature dependence of stable defect

formation in MD simulations: ratio of MD defects to NRT

displacements.

1.11.4.3 In-cascade Clustering of Point

Defects

Among the features visible in the two cascades

shown inFigure 6are a number of small interstitial

clusters For example, the cascade debris from the

1 keV cascade in Figure 6(c) contains only seven

stable interstitials, but five of them (71%) are in

clusters: one di-interstitial and one tri-interstitial

This tendency for point defects to cluster is

charac-teristic of energetic displacement cascades, and it

differentiates neutron and ion irradiation from

typi-cal 0.5 to 1 MeV electron irradiation, which primarily

produces only isolated Frenkel pair defects The

differences between in-cascade vacancy and

intersti-tial clustering discussed below, and the fact that their

migration behavior is also quite different, have a

profound influence on radiation-induced

micro-structural evolution at longer times This impact of

point defect clusters on microstructural evolution is

discussed in detail in Chapter 1.13, Radiation

1.11.4.3.1 Interstitial clustering

The dependence of in-cascade interstitial clustering

on cascade energy is shown inFigure 12for

simula-tion temperatures of 100, 600, and 900 K, where the

average number of interstitials in clusters of size two

or larger at each energy has been divided by the total

number of surviving interstitials in part (a), and by

the number of displaced atoms predicted by the NRT

model for that energy in part (b) The data points and

error bars inFigure 12indicate the mean and

stan-dard error at each energy The error bars can be used

to make two significant comments First, the relative

scatter is much higher at lower energies, which is

similar to the case of defect survival shown in

the standard errors about the mean for interstitial

clustering are greater at each energy than they are

for defect survival

The fact that the interstitial clustering fraction

exhibits greater variability between cascades at a

given energy than does defect survival is essentially

related to the variety of defect configurations that

are possible A given amount of kinetic energy tends

to produce a given number of stable point defects;

this simple observation is embedded in the NRT

model, that is, the number of predicted defects is

linear in the ratio of the energy available to the

energy per defect However, any specific number of

point defects can be arranged in many different ways

At the lowest energies, where relatively few defectsare created, some cascades produce no interstitialclusters and this is primarily responsible for thelarger error bars at these energies The average frac-tion of interstitials in clusters is about 20% of theNRT displacements above 5 keV, which corresponds

to about 60% of the total surviving interstitials.Although it is not possible to discern a systematiceffect of temperature below 10 keV, there is a trendtoward greater clustering with increasing tempera-ture at higher energies This can be more clearly seen

inter-stitials to surviving interinter-stitials is shown, and in thehigh-energy values in Table 2 This effect of tem-perature on interstitial clustering in these adiabaticsimulations is consistent with the observations of Gaoand coworkers77mentioned above, that is, they foundthat the interstitial clustering fraction increases withtemperature

The interstitial cluster size distributions exhibit aconsistent dependence on cascade energy and tem-perature as shown in Figure 13(where a size of 1denotes the single interstitial) The cascade energydependence at 100 K is shown in Figure 13(a),where the size distributions from 10 and 50 keV areincluded The influence of cascade temperature isshown for 10 keV cascades in Figure 13(b), and for

20 keV cascades inFigure 13(c) All interstitial ters larger than size 10 are combined into a single class

clus-in the histograms clus-inFigure 13 The interstitial clustersize distribution shifts to larger sizes as either thecascade energy or temperature increases An increase

in the clustering fraction at the higher temperatures ismost clearly seen as a decrease in the number ofmono-interstitials Comparing Figures 13(b) and13(c) demonstrates that the temperature dependenceincreases as the cascade energy increases The largestinterstitial cluster observed in these simulations wascontained in a 20 keV cascade at 600 K as shown in

interstitials (<111> crowdions), and exhibited siderable mobility via what appeared to be a 1D glide

con-in a<111> direction.64,66Although the number of point defects producedand the fraction of interstitials in clusters was shown

to be relatively independent of neutron energy trum,82 the increase in the number of large clusters

spec-at higher energies suggested thspec-at the in-cascade ter size distributions may exhibit more sensitivity toneutron energy spectrum than did these other para-meters At 100 K, there are no interstitial clusterslarger than 8 for cascade energies of 10 keV or

clus-less Therefore, the fraction of interstitials in clusters

of 10 or more was chosen as an initial parameter

for evaluation of the size distributions This partial

interstitial clustering fraction is shown inFigure 15

As the large clusters are relatively uncommon, the

fraction of interstitials contained in them is

corre-spondingly small This leads to the relatively large

standard errors shown in the figure However, it is

clear that the energy dependence of the formation of

these large clusters is much stronger than simply the

total fraction of interstitials in clusters Infrequentlarge clusters such as the 33-interstitial clustershown in Figure 14 play a significant role in thesharp increase in this clustering fraction observedbetween 100 and 600 K for the 20 keV cascades.One unusual observation reported by Woodingand coworkers60 and Gao and coworkers86 was thatsome of the interstitial clusters exhibited a complex3D morphology rather than collapsing into planardislocation loops which are expected to have lower

0.1 0 0.05 0.1 0.15 0.2

0.25 0.3 0.35 0.4 0.45

Iron cascade simulations

600 K

900 K Mean and standard error: 100 K

0.1 0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

Figure 12 Fraction of surviving interstitials contained in clusters at 100 K; the fraction in (a) is relative to the total number of

MD defects created and in (b) is relative to the NRT displacements.

energy Similar clusters have been seen in materials

such as copper, although they appear to be less

fre-quent in copper.54 The existence of such clusters

has been confirmed with interatomic potentials

that were developed more recently and withab initio

calculations.87 Representative examples of theseclusters are shown in Figure 16, where a ring-likefour-interstitial cluster is shown in (a) and a five-interstitial cluster is shown in (b) Unlike the mobileclusters that are composed of [111] crowdions such

Average interstitial cluster distributions: cascades at 100 K Average interstitial cluster distributions: 10 keV cascades at 100 and 900 K

Average interstitial cluster distributions: 20 keV cascades at 100 and 600 K

Interstitial

Figure 14 Residual defects at 30 ps from a 20 keV

cascade at 600 K containing a 33-interstitial cluster.

Iron cascade simulations

600 K

900 K

0 0.02 0.04 0.06 0.08

Cascade energy (keV)

200

Mean and standard error: 100 K

Figure 15 Cascade energy dependence of interstitials contained in clusters of 10 or more: clustered interstitials divided by NRT displacements.

as the one shown in Figure 14, the SIA clusters

potential for long lifetimes in the microstructure and

may act as nucleation sites for larger interstitial-type

defects Figure 17shows a somewhat larger sessile

cluster containing eight SIA This particular cluster

was examined in detail by searching a large number

of low-order crystallographic projections in an

attempt to find a projection in which it would appear

as a loop Such a projection could not be found

Rather, the cluster was clearly 3D with a single di-,

tri-, and di-interstitial on adjacent, close-packed

(110) planes as shown in the figure The eighth

inter-stitial is a [110] dumbbell that lies perpendicular to

the others and on the left side in Figure 17(a)

three center (101) planes inFigure 17(a)

It is possible that the typical 10–15 ps MD lation was not sufficient for the cluster to reorient andcollapse To examine this possibility, the simulationtime of a 10 keV cascade at 100 K that contained asimilar eight SIA cluster was continued up to 100 ps.Very little cluster restructuring was seen over thetime from 10 to 100 ps In fact, the cluster had coa-lesced into nearly its final configuration by 10 ps Gaoand coworkers86carried out a more systematic inves-tigation of sessile cluster configurations withextended simulations at 300 and 500 K They foundthat many sessile clusters had converted to glissilewithin a few hundred picoseconds, but at least oneeight SIA cluster remained sessile for 500 ps evenafter aging at temperatures up to 1500 K Given theimpact that stable sessile clusters would have onthe longer timescale microstructural evolution as

simu-(b) (a)

4 SIA: 20 keV, 600 K 5 SIA: 20 keV, 600 K

Figure 17 Three-dimensional sessile interstitial cluster in 10 keV, 100 K cascade: (a) [010] projection normal to five adjacent {110} planes, (b–d) projections through three of the {110} planes.

discussed inChapter1.13, Radiation Damage

The-ory, further research is needed to characterize the

long-term evolution of cascade-created point defect

clusters It is significant to point out that the

conver-sion of glissile SIA clusters into sessile clusters has

also been observed For example, in a 20 keV cascade

at 100 K, a glissile eight SIA cluster was trapped and

converted into a sessile nine SIA cluster when it

reacted with a single [110] dumbbell The simulation

was continued for more than 200 ps and the cluster

remained sessile

The mechanism responsible for interstitial

clus-tering has not been fully understood For example,

it has not been possible to determine whether the

motion and agglomeration of individual interstitials

and small interstitial clusters during the cascade

event contributes to the formation of the larger

clus-ters that are observed at the end of the event

Alter-nate clustering mechanisms in the literature include

the suggestion by Diaz de la Rubia and Guinan88that

large clusters could be produced by a loop punching

mechanism Nordlund and coworkers62 proposed

a ‘liquid isolation’ model in which solidification of a

melt zone isolates a region with excess atoms

However, a new mechanism has recently been

elucidated by Calder and coworkers,80 which seems

to explain how both vacancy and interstitial clusters

are formed, particularly the less frequent large

clus-ters Their analysis of cluster formation followed an

investigation of the effects of PKA mass and energy,

which demonstrated that the probability of

produc-ing large vacancy and SIA clusters increases as these

parameters increase.89 The conditions of this study

produced a unique dataset that motivated the effort

to unravel how the clusters were produced They

developed a detailed visualization technique that

enabled them to connect the individual

displace-ments of atoms that resulted in defect formation by

comparing the start and end positions of atoms in the

simulation cell This defined a continuous series of

links between each vacancy and interstitial that were

ultimately produced by a chain of displacements

These chains could be displayed in what are called

lines of ‘spaghetti.’80 Regions of tangled spaghetti

define a volume in which atoms are highly agitated

and a certain fraction of which are displaced Stable

interstitials and interstitial clusters are observed on

the surface in this volume

From their analysis of cascade development and

the final damage state, Calder and coworkers were able

to demonstrate a correlation between the production

of large SIA clusters and a process taking place very

early in the development of a cascade Specifically,they established a direct connection between suchclusters and the formation of a hypersonic recoilatom that passed through the supersonic pressurewave created by the initiation of the cascade Thishighly energetic recoil may create a subcascade and asecondary supersonic shockwave at an appropriatedistance from the primary shockwave In this case,SIA clusters tend to be formed at the point wherethe primary and secondary shockwaves interfere withone another This process is illustrated inFigure 18.80Atoms may be transferred from the primary shock-wave volume into the secondary shockwave volume,creating an interstitial supersaturation in the latter and

a vacancy supersaturation in the former In this case,the mechanism of creating large SIA clusters early

in the cascade process correspondingly leads to theformation of large vacancy clusters by the end ofthe thermal spike phase, that is, after several picose-conds It is notable that the location of the SIA cluster

is determined well before the onset of the thermalspike phase, by about 0.1 ps Calder’s spaghetti analy-sis provides the opportunity for improved definition

of parameters such as cascade volume and energydensity; the interested reader is directed to Calderand coworkers80for more details

1.11.4.3.2 Vacancy clustering

As discussed elsewhere,59,63,65 in-cascade vacancyclustering in iron is quite low (10% of NRT)when a NN criterion for clustering is applied Thiswas identified as one of the differences between ironand copper in the comparison of these two materialsreported by Phythian and coworkers.59 However,when the coordinates of the surviving vacancies

in 10, 20, and 40 keV cascades were analyzed, clearspatial correlations were observed Peaks in thedistributions of vacancy-vacancy separation dis-tances were obtained for the second and fourth NNlocations.64 These radial distributions are shown in

analysis of the vacancy distributions in higher energycascades at 100 and 600 K The peak observed forvacancies in second NN locations is consistent withthe di-vacancy binding energy being greater forsecond NN (0.22 eV) than for first NN (0.09 eV).90The reason for the peak at fourth NN is presumablyrelated to this also since two vacancies that are second

NN to a given vacancy would be fourth NN In tion, work discussed by Djurabekova and coworkers91indicates that there is a small binding energy betweentwo vacancies at the fourth NN distance